Grab the zipfile (portable) of either 32b or 64, instead of the installer, from the download page. Unzip it into a directory where you have full access.

Manaully create your own shortcut to that version, which you should be able to pin or paste wherever your old NPP shortcut was.

This should run with the unzip-directory as your config-storage location. If you want it to use %AppData%\Notepad++ (c:\users\xxxxxx\AppData\Roaming\Notepad++ or similar) for your config/settings, then

Delete the doLocalConf.xml from the unzipped directory (with that file, it will use the unzip-directory for your settings; without the file, it should use AppData)

If you use the portable (zip-file) version, it doesn’t register itself with Windows, so Windows doesn’t automatically list it for choosing with “Open With…” or similar. However, under most versions of Windows, you can select “browse” in the “Open With” dialog, and go find notepad++.exe yourself.

Or, if you are an advanced user, you can edit your association(s) in your registry. (For example, in mine, HKCR\.txt\ uses the (Default) string value with data txtfile, and HKCR\txtfile\Shell\open\command\ uses the (Default) string value of my full path for notepad++.exe. If you don’t understand any of those terms in these parentheses, you probably should be wary of editing your registry yourself.)